The Legendre differential equation is a second-order linear differential equation commonly expressed as (1 - x^2)y'' - 2xy' + n(n + 1)y = 0. It arises in various fields, including physics and engineering, particularly in problems involving spherical symmetry, such as in potential theory and quantum mechanics. Solutions to the Legendre differential equation are known as Legendre polynomials, denoted as P_n(x). These polynomials are orthogonal over the interval [-1, 1] and play a crucial role in approximation theory and numerical analysis. They are also used in solving Laplace's equation in spherical coordinates.